CS61BL. Lecture 5: Graphs Sorting
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1 CS61BL Lecture 5: Graphs Sorting
2 Graphs
3 Graphs Edge Vertex
4 Graphs (Undirected)
5 Graphs (Directed)
6 Graphs (Multigraph)
7 Graphs (Acyclic)
8 Graphs (Cyclic)
9 Graphs (Connected)
10 Graphs (Disconnected)
11 Graphs (Unweighted)
12 Graphs (Weighted)
13 Graph Representation Adjacency Matrix A Matrix (e.g. 2D array) specifying which edges are present in a graph A B C D E F A B D B F C D C E E F
14 Graph Representation Adjacency Matrix A 3 Matrix (e.g. 2D array) specifying which edges are present in a graph A B C D E F 1 A X 3 X X X X B D 1 B X X X 1 1 X -1 C 5 X X X X X F D X X X X X 1 C E 2 E X X 3 X X 2 3 F X X X X X X 5
15 Graph Representation Adjacency List Collection of lists of neighbors Each list specifies a single vertex s neighbors A B C D E F A: [B] B: [D, E] C: [A] D: [F] E: [C, F] F: [ ]
16 Graph Representation Adjacency List Collection of lists of neighbors Each list specifies a single vertex s neighbors A 3 5 B C D E 1 2 F A: [(B, 3)] B: [(D, 1), (E, -1)] C: [(A, 5)] D: [(F, 1)] E: [(C, 3), (F, 2)] F: [ ]
17 Graph Traversal Almost same as tree traversal Keep track of which vertices have been already been visited Pick any vertex to start at instead of root BFS, DFS
18 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E
19 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E A
20 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E
21 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E B
22 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E
23 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E D E
24 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E E
25 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E E F
26 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E F
27 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E F C
28 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E C
29 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E
30 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E
31 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E F
32 Graph Traversal: BFS Example Mark starting vertex as visited and add to queue While queue is not empty: A Remove a vertex from queue Mark its unvisited neighbors as visited and add to queue B D F Head of queue C E
33 Shortest Path Problem Find shortest path from some start vertex s to some end vertex t Google Maps, puzzles, pathing AIs, etc Unweighted graph: use BFS Weighted graph:?
34 Shortest Path Problem Weighted graph? Split each weighted edge into multiple intermediary vertices and edges? Then run BFS? B 1 B 1 C A 3 A C
35 Shortest Path Problem Weighted graph? Split each weighted edge into multiple intermediary vertices and edges? Then run BFS? B 100? 1 C A 300
36 Shortest Path Problem Weighted graph? Split each weighted edge into multiple intermediary vertices and edges? Then run BFS? B ? 1 C A
37 Shortest Path Problem Weighted graph? Split each weighted edge into multiple intermediary vertices and edges? Then run BFS? B ? 1 C A
38 Dijkstra s Algorithm Finds shortest path from start vertex to every other vertex in graph Use priority queue as fringe Priority value of vertex u is shortest path length we ve found from starting vertex s so far
39 Dijkstra s Algorithm Example Add starting vertex s to priority queue w/ priority 0 Add all other vertices with priority While priority queue is not empty: u = vertex dequeued from priority queue For each neighbor v: If v is in priority queue, update v s priority value with minimum of: Current priority Priority of u + weight of edge (u, v)
40 Dijkstra s Algorithm Example Add s to priority queue with priority 0 Add all other vertices with priority Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E?????
41 Dijkstra s Algorithm Example While priority queue is not empty: u = vertex dequeue d from priority queue Priority Queue A B C D E 1 B A 5 2 D 1 9 C 3 4 E Shortest Distances A B C D E 0????
42 Dijkstra s Algorithm Example For each neighbor v, if v is in priority queue, update v s priority value with minimum of: [Current priority, Priority of u + weight of edge (u, v)] Priority Queue A B C D E 1 B A 5 2 D 1 9 C 3 4 E Shortest Distances A B C D E 0????
43 Dijkstra s Algorithm Example For each neighbor v, if v is in priority queue, update v s priority value with minimum of: [Current priority, Priority of u + weight of edge (u, v)] Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0????
44 Dijkstra s Algorithm Example While priority queue is not empty: u = vertex dequeued from priority queue Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0 1???
45 Dijkstra s Algorithm Example For each neighbor v, if v is in priority queue, update v s priority value with minimum of: [Current priority, Priority of u + weight of edge (u, v)] Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0 1???
46 Dijkstra s Algorithm Example For each neighbor v, if v is in priority queue, update v s priority value with minimum of: [Current priority, Priority of u + weight of edge (u, v)] Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0 1???
47 Dijkstra s Algorithm Example While priority queue is not empty: u = vertex dequeued from priority queue Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0 1? 3?
48 Dijkstra s Algorithm Example For each neighbor v, if v is in priority queue, update v s priority value with minimum of: [Current priority, Priority of u + weight of edge (u, v)] Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0 1? 3?
49 Dijkstra s Algorithm Example For each neighbor v, if v is in priority queue, update v s priority value with minimum of: [Current priority, Priority of u + weight of edge (u, v)] Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0 1? 3?
50 Dijkstra s Algorithm Example While priority queue is not empty: u = vertex dequeued from priority queue Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0 1? 3 4
51 Dijkstra s Algorithm Example For each neighbor v, if v is in priority queue, update v s priority value with minimum of: [Current priority, Priority of u + weight of edge (u, v)] Priority Queue A B C D E 1 B A D 1 9 C 3 4 E Shortest Distances A B C D E 0 1? 3 4
52 Dijkstra s Algorithm Example While priority queue is not empty: u = vertex dequeued from priority queue Priority Queue A B C D E B 2 D 1 1 A 5 9 C 3 4 E Shortest Distances A B C D E
53 Dijkstra Example Priority Queue A B C B A 2 C Shortest Distances A B C???
54 Dijkstra Example Priority Queue A B C B -2 3 A 2 C Shortest Distances A B C 0??
55 Dijkstra Example Priority Queue A B C B A 2 C Shortest Distances A B C 0??
56 Dijkstra Example Priority Queue A B C B A 2 C Shortest Distances A B C 0? 2
57 Dijkstra Example Priority Queue A B C B -2 3 A 2 C Shortest Distances A B C 0 3 2
58 Dijkstra Example Priority Queue A B C B -2 3 A 2 C Shortest Distances A B C 0 3 2
59 Another Example Priority Queue A B C B A 2 C Shortest Distances A B C???
60 Another Example Priority Queue A B C B 2 1 A 2 C Shortest Distances A B C 0??
61 Another Example Priority Queue A B C B A 2 C Shortest Distances A B C 0??
62 Another Example Priority Queue A B C B 2 1 A 2 C Shortest Distances A B C 0 1?
63 Another Example Priority Queue A B C B A 2 C Shortest Distances A B C 0 1?
64 Another Example Priority Queue A B C B 2 1 A 2 C Shortest Distances A B C 0 1 3
65 Another Example Priority Queue A B C B 2 1 A 2 C Shortest Distances A B C Cycles are fine
66 Why do we care about graphs? Facebook, LinkedIn Maps (Google Maps, GPS) The internet Puzzle configurations (Rubik s cube, project 3)
67 Final Exam Information Next Wednesday (8/13), 3-6pm Last name A-H, Q-Z: 2050 VLSB Last name J-P: 100 GPB All lab and quiz material up to and including this Friday s (balanced search trees) Review Session Sunday (8/10), 1-3pm in 100 GPB
68 Sorting Given a collection of elements, sort them from smallest to largest
69 Terminology: Stable Elements with the same value are ordered the same before and after sorting Original: Sorted (stable): Sorted (unstable):
70 Terminology: In-place Only uses O(1) additional space E.g. doesn t copy all contents of input array into a new array Used to describe any algorithm, not just sorting algorithms
71 Selection Sort for i = 0, 1,..., (n-1): Search positions i to(n-1)for smallest element Swap with element at position i Stable, in-place Best / average / worst case running time: O(n 2 ) for a list with n elements
72 Selection Sort Example Current index Smallest found this iteration Already sorted
73 Bubble Sort for i = 0, 1,..., (n-1): Go through the first n-i elements Swap pairs of adjacent elements that are out of order Stable, in-place Average / worst case running time: O n 2 for a list with n elements Best case running time: O n with optimization
74 Bubble Sort Example
75 Insertion Sort for i = 0, 1,..., (n-1): Take element at index i Insert it into the correct location in the sorted sublist consisting of the first i+1 elements Stable, in-place Average / worst case running time: O n 2 list with n elements Best case running time: O n for a
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120 Faster Sorting Divide and Conquer: Solve large problem by splitting it into smaller instances of same problem
121 Heap Sort (From last lecture) Create heap Remove smallest element from heap, one at a time NOT Stable In-place* Best / average / worst case running time: O(n log n) for a list with n elements
122 In-place Heap Sort X
123 In-place Heap Sort X
124 In-place Heap Sort X
125 In-place Heap Sort X
126 In-place Heap Sort X
127 In-place Heap Sort Etc
128 In-place Heap Sort X
129 Merge Sort Base case: list of 1 element is already sorted Recursively sort left and right halves Merge Stable* Can be done in-place (not efficient) Best / average / worst case running time: O(n log n) for a list with n elements
130 Merge Sort Running Time
131 Merge Sort Running Time
132 Merge Sort Running Time 1 merge: lists of n 2 elements each 2 merges: lists of n 4 elements each n 2 merges: lists of 1 element each
133 Merge Sort Running Time 1 merge: lists of n elements each 2 2 merges: lists of n elements each 4 n 2 merges: lists of 1 element each O(n) O(n) O(n) log n depth
134 Merge Sort Running Time 1 merge: lists of n elements each 2 2 merges: lists of n elements each 4 n 2 merges: lists of 1 element each O(n) O(n) O(n) log n depth O(n log n)
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155 Quick Sort Base case: list of 1 element is already sorted Choose one of the list elements as the pivot Partition list into elements that are less than pivot and elements that are greater than pivot Recursively do this on both partitions Stable, in-place Best / average case running time: O(n log n) where n is number of elements in input list Worst case running time: O(n 2 )
156 Quick Sort Running Time Pivot Pivot Pivot
157 Quick Sort Running Time Pivot Pivot
158 Quick Sort Running Time All partitioning at each depth level: O(n) time Number of depth levels: O(log n) to O(n) Total running time: O(n log n) to O(n 2 )
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175 Sorting in practice Base case: use insertion sort on smaller lists Quick sort is fastest on randomized lists Choosing a pivot (options): Choose first element of list Calculate median Pick 3 elements at random and take median
176 Java s Arrays.sort Dual-pivot quicksort! Vladimir Yaroslavskiy, 2009 Used in Java s Arrays.sort starting with Java 7 (2011)
177 Comparison Sorts Compares elements Given items a and b: a > b? a < b? (a == b?) All sorting algorithms we ve learned so far Selection, bubble, insertion, merge, heap, quick
178 Comparison Sorts: Number of Comparisons (Lower Bound) Each comparison distinguishes between 2 possibilities f(n) comparisons 2 f n list permutations n elements: n! possible list permutations 2 f n n! f n log 2 (n!)
179 Comparison Sorts Lower bound on number of comparisons: log 2 (n!) Want to show: log 2 (n!) Ω(n log 2 n)
180 Want to show: log 2 (n!) Ω(n log 2 n) log 2 (n!) = log 2 n + log 2 n log 2 2 n > log 2 n + log 2 n log 2 2 > log 2 n 2 + log 2 n log 2 n 2 = n 2 log 2 n 2
181 Want to show: log 2 (n!) Ω(n log 2 n) log 2 n! > n log n Θ(n log 2 ) 2 Θ(n log 2 n n log 2 2) Θ(n log 2 n) log 2 (n!) Ω(n log 2 n) n Lower bound on number of comparisons: Ω(n log 2 n)
182 Comparison Sorts Lower bound on number of comparisons: Ω(n log 2 n)
183 Non-comparison Sorts We can do better
184 Counting Sort Input: Large unsorted list of n integers in the range [0, k] (hopefully k n) Solution: Create an int array of size k Traverse input list Increment int at array position i whenever you come across i
185 Counting Sort Example Input list Counter array
186 Counting Sort Example Input list Counter array
187 Counting Sort Example Input list Counter array
188 Counting Sort Example Input list Counter array
189 Counting Sort Example Input list Counter array
190 Counting Sort Example Input list Counter array
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192 Counting Sort Example Input list Counter array
193 Counting Sort Example Output: Counter array
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201 Counting Sort Example Output: Counter array
202 Counting Sort Best / average / worst case running time: O n + k Stable? N/A In-place? No
203 Radix Sort (version 1) Base case: list of only 1 element is already sorted Put elements into buckets according to most significant digit Recursively sort each bucket on next most significant digit (and so forth)
204 Radix Sort v1 Example Input list: 49, 26, 32, 39, 65, 61, 33, 31
205 Radix Sort v1 Example Input list: 49, 26, 32, 39, 61, 65, 33, 21 Put elements into buckets according to most significant digit
206 Radix Sort v1 Example Input list: 49, 26, 32, 39, 61, 65, 33, 21 Recursively sort on next most significant digit(s)
207 Radix Sort v1 Example Input list: 49, 26, 32, 39, 61, 65, 33, 21 Output sorted list: 21, 26, 32, 33, 39, 49, 61,
208 Radix Sort (version 2) Base case: list of size 1 is already sorted Put elements into buckets according to least significant digit Output all of the elements to a list: In bucket order In the order they were added to each bucket Repeat with next digit until you reach the most significant digit
209 Radix Sort v2 Example Input list: 49, 26, 32, 39, 61, 65, 33, 21
210 Radix Sort v2 Example Input list: 49, 26, 32, 39, 61, 65, 33, 21 Put elements into buckets according to least significant digit
211 Radix Sort v2 Example Output all of the elements to a list: In bucket order In the order they were added to each bucket Output: 61, 21, 32, 33, 65, 26, 49,
212 Radix Sort v2 Example Input: 61, 21, 32, 33, 65, 26, 49, 39 Repeat with next digit
213 Radix Sort v2 Example Input: 61, 21, 32, 33, 65, 26, 49, 39 Output: 21, 26, 32, 33, 39, 49, 61,
214 Radix Sort Running Time Best / average / worst case: O( n + b k) n: Number of elements in input list b: Number of buckets k: Number of digits of largest number Usually, b n, so running time can be reduced to: O(nk) If k is also small (basically a constant), running time can be reduced to: O n Sorting Java ints: b=2, k = 32
215 Good luck next week.
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